Abstract
We consider the obstacle problem for the Gauss curvature flow with an exponent \(\alpha \). Under the assumption that both the obstacle and the initial hypersurface are strictly convex closed hypersurfaces and that the obstacle is enclosed by the initial hypersurface, uniform estimates are obtained for several curvatures via a penalty method. We also prove that when the hypersurface is two dimensional with \(0<\alpha \le 1\), the solution of the Gauss curvature flow with an obstacle exists for all time with bounded principal curvatures \(\{\lambda _i\}\), where the upper bound is uniform, and the lower bound depends on the distance from the free boundary. Moreover, we show that there exists a finite time \(T_*\) after which the solution becomes stationary in the same shape as the obstacle.
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Communicated by P. Topping.
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The authors would like to thank the anonymous referee for carefully reading the manuscript and for valuable comments that contributed to improve the presentation. Ki-Ahm Lee has been supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1701-03, and holds a joint appointment with the Research Institute of Mathematics of Seoul National University. Taehun Lee was supported by National Research Foundation of Korea Grant (NRF-2014H1A2A1018664) funded by the Korea government and has been supported in part by a KIAS Individual Grant (MG079501) at Korea Institute for Advanced Study.
